Polygons of constant width and height, with an odd number of sides greater than or equal to three, also known as Reuleaux polygons (named after Franz Reuleaux, a German engineer who lived from 1829-1905), have unusual and useful properties which would be understood to one of skill in the art. For example, manhole covers are almost always round, because if they were any other common shape such as a rectangle, a triangle, or an oval, having one side smaller than the other means they would easily fall into their own manhole during installation or if dislodge by a passing vehicle, Reuleaux polygon shapes, such as a Reuleaux triangle, or pentagon, improve on the circle; not only will they not fall into a corresponding manhole, but they will also not roll away almost indefinitely as a circular-shaped manhole cover would. Reuleaux-shaped manhole covers, unlike circle-shaped manhole covers, also will not rotate when installed, so that warnings, text or images written thereon will stay oriented the same way as when they were installed. Further, and importantly as security is a growing concern for large public events such as marathons, Reuleaux-shaped manhole covers are difficult to counterfeit or tamper with, especially when welded shut.
Because of their beneficial properties, Reuleaux shapes have also been used for coins in certain countries, especially in the Commonwealth of England, and most often as Reuleaux heptagons. They are difficult to counterfeit, and due to their constant width, roll easily in vending machines. Reuleaux shapes have also been used in pencils and drill bit inserts.
Reuleaux shapes also have a constant height as they are rolled along a flat surface, and thus a flat object, such as a ruler laid atop one or more identical rolling Reuleaux shapes will, translate horizontally but not vertically. The center of the Reuleaux shape will, however, move up and down.
Reuleaux shapes can be made by subtractive manufacturing, that is, using equipment such as saws, drills, milling machines, lathes and CNC machines to take away extra material from a larger object until it is in a Reuleaux shape. The extra material that was purchased becomes waste, and has to be discarded or recycled at an additional cost.
Reuleaux shapes can also be made by additive manufacturing, where parts are assembled, welded, glued or bolted on. However, this is ineffective particularly for certain material that will not fuse with separately created-pieces of like material (such as ceramics and certain plastics including polypropylene).
Reuleaux shapes can also be made by casting or 3D printing, but these are expensive options.
Accordingly, it would be advantageous for there to be systems and methods for generating Reuleaux shapes, without the need for specialized molds, machining, carving, stamping, or tooling, all of which would add to the manufacturing cost.